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How would I determine if the following series is absolutely convergent, conditionally convergent or divergent?

$\sum\limits_{i=1}^n$$\sqrt[n]{2}+1$

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  • $\begingroup$ Firstly, this series is not geometric. Secondly, what have you tried? $\endgroup$ – James Nov 15 '15 at 2:07
  • $\begingroup$ I tried to begin with the root test but got confused pretty quick $\endgroup$ – Glenn2329 Nov 15 '15 at 2:08
  • $\begingroup$ Well the root test is just going to introduce another root, so, you will have a root of a root, which isn't really any simpler. The root test is good when you have $n$th powers, not $n$th roots. For a rough heuristic as to orders to test series convergence, I suggest doing the divergence test first i.e. does the *sequence* $(\sqrt[n]{2} + 1)$ tend to $0$ as $n\rightarrow\infty$? $\endgroup$ – James Nov 15 '15 at 2:11
  • $\begingroup$ Do you mean $\sum_{n=1}^\infty(\sqrt[n]2+1)$? $\endgroup$ – Tim Raczkowski Nov 15 '15 at 2:24
  • $\begingroup$ yes thats what i meant $\endgroup$ – Glenn2329 Nov 15 '15 at 2:31
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Notice that $\lim_{n\rightarrow\infty}\sqrt[n]2+1=1+1=2\neq0$. Hence the series diverges.

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